October 1965 Stanley L. Rosenthal 605
SOMEPRELIMINARYTHEORETICALCONSIDERATIONSOFTROPOSPHERIC
WAVEMOTIONS IN EQUATORIALLATITUDES
STANLEY L. ROSENTHAL
National Hurricane Research Laboratory, Environmental Science Services Administration, Miami, Fla.
ABSTRACT
Wave solutions t o t h e linearized, quasi-hydrostatic equations for adiabatic, nonviscous flow on an equatorially
oriented beta plane are obtained. The basic current is assumed t o b e zonal and invariant in both space and time.
Only solutions for which the meridional wind component is symmetric with respect to the equator are considered.
Disturbances with wavelengths on the order of 103 km. are found to be very nearly nondivergent. The solutions
show the meridional wind component to be very nearly geostrophic even at very low latitudes. The perturbation of
the zonal wind, however, is highly ageostrophic a t t h e very low latitudes and significantly ageostrophic even in
subtropical latitudes.
1. INTRODUCTION
Charney's [3] scale analysis predicts that nonviscous,
adiabatic, synoptic-scale atmospheric motions near the
equator should be very nearly nondivergent and, there-
fore, adequately described by the conservation of absolute
vorticity. In the present study, such motions are examined
in terms of certain wave solutions obtained from the linear-
ized , quasi-hydrostatic equations for adiabatic, nonviscous
flow on a beta plane.
Aside from the assumptions cited above, the basic cur-
rent is taken to be zonal and invariant in both space and
t,ime. The discussion, therefore, pertains only to existing
perturbations and provides no information concerning the
manner in which they originate. It is not claimed that the
theory is applicable to observed equatorial wave motions
such as those described by Palmer [6]. It is quite likely
that the latter are significantly affected by the release of
latent heat and by quasi-barotropic instabilities due to
meridional shears of the basic current. However, as will
be seen later, there is a fairly reasonable superficial
resemblancebetween the structure o f the observed and
theoretical waves.
Previous theoretical studies of disturbances in equatorial
latitudes have assumed the flow to be nondivergent [lo] or
have simplified the problem through specification of one of
the velocity components [7, 111. Freeman and Graves [4]
make both of these simplifications. Neither is included in
the model developed below. On the other hand, Rosenthal
[7] and Sherman [I 11 allowed the basic current to vary with
latitude; this is not done here.
-+u* -+a* -+w* -+pyu*+-=o dv* dv* du* dv* ** dt dx by dp by ' (2)
and
-="-+-). dW* du* dv*
32) dx dy
Here, t is time, x is zonal distance, y is meridional distance
measured positive northward from the equator, p is pres-
sure, u* is the zonal wind component, v* is the meridional
wind component, W* is the p-system vertical motion, 4*
is the geopotential of isobaric surfaces,
is the static stability, B=-"aconstant andf is the Coriolis
parameter.
d j
a?/
The dependent variables are written,
u*=U+u(x, y, p, t ), U=a constant (5 )
v*=v(z, Y, Pl t ) (6 )
o*=w(x, Y, p, t> (7)
+*=ay/, P)+4(X, Y, P, t> (8)
and
The variables, u, v, W, 4 are perturbation quantities.
Since the base state must satisfy the governing equations,
2. SOLUTION OF THEPERTURBATIONEQUATIONS
The equations whichgovern nonviscous, adiabatic,
we find
quasi-hydrostatic, beta plane flow are, and, upon integration,
du* du* du*
at ax by dP bX pyv*+-=o, (1) "fu* -+v* "$-w*" &*
606 MONTHLY WEATHER REVIEW
Substitution of (5 ), (6), (7), and (8) into (l), (2), (3), and
(4), and utilization of (9) and (10) yields, after lineariza- where
tion by the usual technique,
~+u--pyv+-=o bU a4 From (26), (27), (28), and (29), we obtain
~d2&+ --X -+- Q=o
bt b x bX (11)
~+u-+pyu+-=o bV * dXZ (i ):
(30)
at ax all (I2) where
--+u-+bw=O 3% b24
bpbt bpbx
and
(14) Equation (30) is a special case of the confluent hyper-
geometric equation ([5] p. 96)
where
s -+(b-~) d2T d T --~T=o
ds2 ds
is assumed constant. If, a t t=O, we require v to attain its whose general solution may be written
maximum value a t x=O, p=po, then the system (1 1),
(12), (13), and (14) has solutions of the form T=K,M(a, b, s)+Kzsl-bM(a-b+l, 2--6, s)
u=A(y) sin k(x-ct) cos m(p-po) (16) where
v=B(y) cos k(x-ct) cos m(p-po) (17) M(a, b, s)=1+--.-+"-+ b l! b(b+l) 2! . . .
+=H(y) sin k(x-ct) cos m(p-po) (la) The general solution to (30) is then
w=W(y) cos k(x-ct) sin m(p-po) (19)
where po=lOOO mb., k=2r/L, L is the wavelength, c is
a s a(a+1) s2
the wave speed, and
m = mJp0 (20) read
By use of (27) and (28), equation (32) may be arranged to
n is an integer equal to the number of surfaces of nondi-
vergence. From (19) and (20), w=O a t p=O and p=po. B =K ,~ 2y M(", Substitution of (1 6 ), (1 7 ), (18), and (19) into (l l ), (12), 2 2 7
8u2 "
(13), and (14) gives
kAA-fiyB+kH=O '
-
and
mW=- kA+-
( 3 wh%re
A=U-C
By elimination of variables between (21), (22), (23), and
(24), we obtain the following equation for B:
(p- k2A) ( &A2- Z )]
Fp2A B=O (26)
The following new variables are introduced,
The parameter a is an eigenvalue of the problem and
must be determined from side conditions imposed upon the
motion. If we restrict ourselves to flows in which u is
symmetric about the equator (this is the case, for example,
with the Palmer waves), then &-LO. If it is further
required that v vanish a t
y =f y ,
then a= a* where
In general, numerical methods are required if (35) is to be
solved for a*.
A simpler solution is obtained when the condition
v( & ym) =O is replaced by the less stringent restriction that
v decay with distance from the equator. This can be
achieved by selecting a=O in which case
October 1965 Stanley L. Rosenthal 607
and, from (31),
(37)
The roots of (37) are'
A,=-? 2 [1+(1+8>11']
A3= [(l+gJl2-1]
(39)
Table 1 lists some values of A I , A 2 , and A3 (Z=3 m.t.s.
units; a reasonable value for the tropical mid-troposphere).
AI is a pure internal gravity wave which propagates rap-
idly westward relative to the basic current. A2 is an
inertia-gravity wave which propagates rapidly eastward
relative to the basic current. A3 is also an inertia-gravity
wave. However, its propagation relative to the basic cur-
rent is westward at approximately the Rossby rate of
AND=p/k2 (it is shown in the appendix that A3 reduces to
AND if the motion studied here is constrained to be nondi-
vergent). According to Palmer's [6] observations, the
motion of equatorial waves relative to the basic current
is small and, therefore, we assume A3 to be the meteoro-
logically significant root.
Equation (40) may be written,
When the basic current is easterly (E= - U>O),
CND= - E-AND
and
(43)
The disturbances, therefore, propagate westward at a
speed which exceeds that of the basic current but which
is less than would be the case for nondivergent flow. The
distribution of convergence and divergence must then
be such that westward relative motion is retarded. This
implies convergence to the east of zones of cyclonic
relative vorticity and divergence to the west of such
regions. The reverse is true of zones of anticyclonic
relative vorticity and the rule is equally valid in both the
Northern and Southern Hemispheres and for easterly
and westerly basic currents.
TABLE 1.-Roots of the frequency equation (37) as given by equations (38), (39), and (40). Values are given in m. see. -I. n=l corre- sponds to one surface of nondivergence; n=9 corresponds to two
surfaces of nondiuergence. The static stahility is taken as 3 m.t.s. units. AN~=j3/19 i s the value of A appropriate to nondivergent motion.
n=l 7k= 2
L (km.) 1 AND I AI Az A3
1
A2 Aa
2OOo
1000
2.3 5.2 3000
0.55 -28.2 27.5 0.55 -55.6 55.0 0.58
10.5 -38.2 27.5 7.3 -35.0 27.5 11.8 -66.7 55.0 14.4 5000
4.4
8.0 -63.0 55.0 -32.0 27.5 9.3 4000
2.2 -29.8 27.5 4.7 -59.7 55.0 2.2 -57.2 55.0
From table 1, it is clear that A3 departs more and more
from AND as the wavelength increases. The model
convergence and divergence is then of greater signi-
ficance at the longer wavelengths. This is similar to the
so-called "barotropic divergence" found in barotropic,
quasi-geostrophic models which allow non-zero vertical
integrals of the divergence [l, 2, 8, 9, 12, etc.] and is
basically different from the divergence discussed in
Charney's [3] scale analysis of low-latitude atmospheric
motions. The latter has magnitude
r 73
(44)
where U is the characteristic velocity, L is the char-
acteristic scale, g the acceleration of gravity, e the
potential temperature, and H the scale height. This
divergence is seen to decrease with increasing wavelength.
and (36), it is a simple matter to complete the solutions
for u, v, C$ and w. These are
By use of (161, (17), (181, 0 9 ), (211, (22), (231, (241,
Eu2
and
B U Z
"
W=m7(A3+7) 'A3Y voe 27 cos k(z-ct) sin m(p-po). (48)
3. DISCUSSION OF THE SOLUTIONS
The divergence may be calculated from (48) as
-bw/bp. The amplitude of the divergence is then
8u2
(49)
over, when A=y, (21), (22), (23). and (24) may be solved to yield B=o&xp(-Py*/D) thus
I Thederivation ofequation (30)required division by k(;-m2A2)=krn? (??-A?). 1Iow-
showing that (36) is valid even for this casc. @I reaches its maximum magnitude a t
608 MONTHLY Vol. 93, No. 10
y= .(;)
1/2
hence,
Table 2 gives I@[max as a function of wavelength for
a=3 m.t.s., v0=5 m. sec.", n=l and n=2. The values
are found to be extremely small. Although, as mentioned
in the previous section, the divergence found in this
model is basically different from that of Charney's [3]
discussion, our results do confirm Charney's conclusion
that it is necessary to include precipitation and the release
of latent heat in order to obtain realistic vertical motion
and divergence at equatorial latitudes. Table 2 also
shows the divergence increases in magnitude as the wave-
length increases. This confirms our conclusions based on
a discussion of the frequency equation in the previous
section. It also emphasizes the different nature of the
divergence of this model and that of Charney's model
(also mentioned in section 2).
-
from (45) and (46). This quantity reaches its.maximum
amplitude
at the equator. Values of I P Jmax are given by table 3;
the ratio I@)lmax/l { lmax is shown by table 4. Wefind
the relative vorticity to have an order of magnitude which
is meteorologically significant and which is large com-
pared to that of the divergence. Relative to the Palmer
waves [6], the model vorticity is of the correct order of
magnitude. However, the Palmer waves, which typically
have a wavelength of about 2000 km., show divergences
on the order ofsec.". Hence, the model divergence
is small compared to that of the Palmer waves. The
latter, therefore, should probably be attributed to the
release of latent heat in regions of organized convection
associated with the waves.
It is of interest to examine the extent to which the
perturbation-velocity components are in geostrophic equi-
librium. For this purpose, we define the following ratios:
Ibu I T7bul
It is clear that small values of R o z and Ro, correspond to
near geostrophic flow. Large values indicate highly
ageostrophic motion. Roz is independent of latitude while
Ro, does not vary with wavelength. From table 5, we see
that Roz is quite small for the shorter wavelengths. For
these wavelengths, therefore, the symmetrical (v) com-
TABLE 2.-Values cf the maximu_m magnitude of the divergence as
computed from equation (60). a=S m.t.s. and v0=6 m. set.".
Values aregiven in units of set.".
L (km.) I n=l 1 n=2
~~
TABLE 3.-Values of the maximum- magnitude of the relative vorticity
computed from equation (51). a=S m.t.s. and 00=5 m. set.".
Values are given in units of set."
L(km.) I n=l 1 n=2
1000 .......___
2ooo ...._._.._
5.OXlO-5 5.OXlO-3
1.6XlO-5 1.4XlO-3 4ooo ....._....
1.9XlO-5 1.8XlO-5 3ooo ______....
2.7XlO-5 2.6XlO-3
5 m ._____.... 1 .4 ~1 ~ 1 .2 ~1 0 -5
TABLE 4.- Values of the ratio !!%%?. a=S m.1.s. I I lmsx
L(km.) I n=l 1 n=2
2wo _____.....
3000 ...._._...
7.OXlO-3 3.OXlO-3
5.6XlO-2 2.9XlO-2 5ooo ._._._....
3.7XlO-2 1.8XlO-2 4wo .___......
2.OX10-2 8.9XlO-3
1000 ._.__..... 1 .1 ~1 ~3 4 .0 ~1 0 -4
ponent of the perturbation motion is very nearly in
geostrophic equilibrium even at very low latitudes. The
behavior of the asymmetrical (u) perturbation component
is quite different. Ro, tends to infinity as the equator is
approached. Table 6 shows that u is markedly ageo-
strophic even in subtropical latitudes.
The fact that the meridional wind component of our
model is very nearly in geostrophic equilibrium helps to
explain the different behaviors of the divergence of this
model and that of Charney 131. According to Charney's
theory, the horizontal acceleration balances the horizontal
pressure-gradient force. The horizontal temperature
gradients needed to estimate the vertical motion and
divergence are, therefore, computed from vertical shears of
the horizontal acceleration. In the model developed here,
i34/& (the only component of Vcp which affects w in the
thermodynamic equation) is given, approximately, by the
Coriolis term, Pyv, and not by the horizontal acceleration.
It is to be noted that part of this discrepency stems
directly from the decision to treat only the syrnIqetrica1
part of (33). It is likely that the asymmetrical solution
would give results more like those of Charney.
The role of divergence in the vorticity equation within
the framework of our model may be determined by a,
study of the ratio
f (V *V )
P V
(54)
October 1965 Stanley L. Rosenthal 609
TARLE 5.-Values of RQz computed from equation (5.2). ~=3 s m ~t .s . units.
L(km.) n=2 n=l
1000"- """_ 2ooo- - - - - - - - - -
0.01 0.02
0.04 0.07
3000 ""_ _" - - 0. os 0. 14
4000"- "_"" 5000- "
"_" - -
0.13
0. 18 0.28
0.21
TARLE 6.-Values of RQ, computed from equation (53). 2 =3 m.1.s. units.
Lat. (deg.) I n=l 1 n=2
195 49.1 21.8 12.2
7.40
1.95 0.87
0.44
0.31 0.22
97.5 24.5 10.9
3.70
6. 10
0.98 0.44
0.22
0. 16 0. 11
TABLE 7.-Values of the ratio computed from equation
(64).a=S m.f.s. units
Lat. (deg.) 1 L=2ooo km., n=l 1 L=5ooo km., n=2
where j (0 'V) is the amplitude of j "I-- and
pv is the amplitude of pV. Values of this ratio for
L=2000 km., n =l and L=5000 km., n=2 are listed
by table 7. With L=2000 km., n=1, we find, as
predicted by Charney [3], that the divergence term in the
vorticity equation is negligible at very low latitudes.
On the other hand, with L=5000 km., n=2, this term
becomes significant within 5" to 10" lat. of the equator.
We now turn to a description of some synoptic aspects
of the theoretical disturbances. Since the motion is very
nearly nondivergent, it is convenient and appropriate to
introduce a stream function. From the Helmholtz
theorem,
(:: Z;)
u=--+- b* bx
b y bx
and
(55)
where fi is the stream function for the perturbation motion
and x is the corresponding velocity potential. Unfortu-
nately, when u and v are given by (45) and (46), J. and X
cannot be obtained in closed form. Since we require +
only for the purpose of presenting a pictorial representa-
tion of the flow pattern and since the flow is very nearly
nondivergent, the following approximate technique will
- 0
I
"-
I
t l -
f7
14
...
4
I/
1
Nor11
1
FIGURE 1.-Perturbation-stream function computed from equation
(57). Isopleths are labeled in units of lo5 m.2 set.", ~=3 m.t.s.,
L=2000 km., u0=5 rn. set.", t=O, p =p o , n =l .
-
suffice.We neglect the potential flow in (55) and (56),
evaluate u and v from (45) and (46), respectively, and
integrate the resulting expressions for b+/bx and b$/by.
This yields two values for # which are only approximately
equal due to the neglect of bx/bx and ax/dy. A final
estimate is obtained by averaging the two approximations.
The result is,
Figure 1 shows the +pattern for a=3 m.t.s., v0-5 m.
sec.", 1;=2000 km., n=l, t =O , and p=lOOO mb. The
motion consists of alternating, north-south elongated cells
of clockwise and counterclockwise circulations each of
which is centered on the equator. The clockwise cells are
cyclonic in the Southern Hemisphere and anticyclonic in
the Northern Hemisphere. The reverse is true of the
counterclockwise cells. Relative to the field of perturba-
tion-pressure height (computed from (47) and shown by
fig. 2), cyclonic perturbation motion is associated with low
perturbation pressure-height and the reverse is true of
anticyclonic perturbation motion. The field of perturba-
tion pressure-height itself is composed of alternating Highs
61 0 MONTHLYWEATHERREVIEW Vol. 93, No. 10
-u f\
2 2
- . 0 -I
I
I \
-0-
0
\'
4 0 1
-NO,
- 2 0
- 15'
- 100
- 54
2 2
- 5'
- 10-
-15'
- 20.
-sout
FIGURE 2.-Perturbation pressure-height computed from equation FIGURE 3.-Total-stream function obtained by adding to figure 1
(47). Isopleths are lfbeled in units of meters. Parameters are the stream function for a constant easterly current of 7.5 m. set.".
the same as for figure 1. Isopleths are labeled in units of lo5 m.* set.". Parameters
as for figure 1.
and Lows centered asymmetrically at some distance from
the equator. The amplitude of the pressure-height per-
turbation is only a few meters and, therefore, if waves of
this type were to exist in the real atmosphere, the varia-
bility of pressure-height associated with them would prob-
ably go undetected. The total-stream function and
pressure-height (figs. 3 and 4, respectively), which consists
of the sums of the perturbation and base-state quantities
(with v= -7.5 m. sec.") shows some resemblance to the
streamlines and pressure patterns of Palmer's [6] empirical
model (see his figs. 8 and 9).
The 1000-mb. fields of perturbation-relative vorticity
and divergence are shown, respectively, by figures 5 and 6.
Positive relative vorticity corresponds to counterclockwise
turning and, hence, to cyclonic relative vorticity in the
Northern Hemisphere and anticyclonic relative vorticity
in the Southern Hemisphere. The reverse is true for
negative relative vorticity. Thus, as would be expected
from our comparison of the perturbation-stream function
with the perturbation pressure-height, cyclonic relative
vorticity is associated with low perturbation pressure-
height and the reverse is true for anticyclonic relative
vorticity.
Everywhere, except at the equator itself, convergence
occurs to the east of cyclonic relative vorticity and diver-
gence occurs to the east of anticyclonic relative vorticity
(fig. 6). This is consistent with the distribution of con-
vergence and divergence previously deduced from the
-frequency equation. The spatial distribution of conver-
gence and divergence is in good agreement with that of
Palmer's empirical model ([6] fig. IO), however the mag-
nitudes given by the theory are far too small.
4. CONCLUSIONS
In equatorial lat.it.udes, nonviscous, adiabatic pertur-
bations superimposed upon an invariant basic current
are very nearly nondivergent provided that the wave-
length is of the order lo3 km. and the meridional wind
component is symmetric with respect to t>he equator.
Also, the meridional wind componenb will be very nearly
geostrophic even a t very low latitudes. In these cir-
cumstances, the zonal component of the perturbation
wind will be asymmetric with respect to the equator and
highly ageostrophic in equatorial latitudes. The order
of magnitude of the relative vorticity obtained from the
October 1965 Stanley L. Rosenthal 61 1
5+ 45*
5' t 1"'
FIGURE 4.-Total pressure-height obtained by adding to figure 2
the pressure-height computed from equation (10) for a basic
easterly current of 7.5 m. set.". Isopleths are labeled in units
of meters. Parameters as for figure 1 .
theory is in agreement with that of observed equatorial
disturba.nces as described by Palmer [6]. The order of
magnit>ude of the divergence given by the model is quite
small compared to that. found empirically by Palmer [6].
This result, taken toget,her with Charney's [3] work, would
seem to indicate that' models of equatorial flow must ta.ke,
into account the latent heating produced by organized
convection in order to provide realistic vertical motions
and divergences.
The amplitude of the geopotential perturbation given
by the theory is extremely small; with present standards
of observation in equatorial lat,itudes, i t would probably
be undetectable. As is the case in higher latitudes, the
modelshows cyclonic flow to be associated with low
geopotential and the reverse to be true of anticyclonic
flow.
APPENDIX
lnspection of (23) and (24) shows that the solution for
nondivergent motion can be obt,ained by allowing a to tend
toward infinity while H, A, k, and m remain finite. In this
case, (40) yields
786-520 0 -6 6 --5
15'-
10'-
5' -
UDllll -
5' -
10' -
15'-
15
\I 5
-15
-15'
0 -
- 204
15*
- 10'
- 5'
,i 5,
E W t I
- 5'
1
'I J
FIGURE 5.-Relative vorticity computed from ecruations (45) and
(46). Positive values indicate counterclockwise rotation. Iso-
pleths are labeled in units of 10-6 ser.". Parameters as for
figure 1.
A3= lim
Y+-
-1 2 [(1 +3 2 -q
1
L Y l
or, if 1'Hospital's rule is employed,
From (36) the nondivergent solution for B is
B=v~ (A.3)
provided that, y is not allowed to approach infinity. From
(24) 1
A=O (A .4)
for nondivergent motion. From (21), the nondivergent
value of H is
PY
k (A -5) H=- DO
Equations (A.3) and (A.4) are exactly the assumptions
61 2 MONTHLY WEATHER REVIEW Vol. 93, No. 10
!
i I j
‘I
I
!
i -i! -
i . P5,.
i
”
5 -25
Nor1
- 100
-50
’\ -54
8 5
”0 EWDI
25
,/’
,’ -5’
10’
15’
1
PI~URE 6,”L)ivergence computed from equations (45) and (46).
Isopleths are labeled in units of 109 sec.”. Parameters as for
figure 1.
employed by Rossby [SI. Furthermore, (A.2), (A.3),
(A.4), and (A.5) correspond precisery to Freeman and
Graves’ [4] recent theory of equatorial Rossby waves.
Alternately, (A.2), (A.3), (A.4), and (A.5) may be
obtained by eliminating H between (21) and (22) to
form the “vorticity equation” and then imposing the
restriction
dB
dY - kA+--O
The latter constrains the motion to be nondivergent.
Upon executing this sequence of operations, equation
(A.7) is obtained.
2 A=O corresponds to a trivial solution in the nondivergent cdsc
The solution for sgmmetricd B is
B=v~ COS
~
(2z+1)?r y, 1=an integer (A.s) 2Yw
provided that
(2Z+1)2~2-k2A-~
4&
“
A (A.9)
If, as was done in the derivations of (36), we require B
to have no zeroes over the range of y, then yw must
approach infinity. In this case, (A.8) reduces to (A.3),
(A.9) gives (A.2), (A.6) gives (A.4), and (A.5) may be
obtained from (21).
REFERENCES
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of Using the Balance Equation for Three-Dimensional Flow,”
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Journal of the Atmospheric Sciences, vol. 20, No. 6, Nov. 1963,
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in the Tropics,” in Interim Report, No. 1, Contract DA 28-043
AMC-O0173(E), National Engineering Science Co., 1964,
106 pp.
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Easterly Waves,” Journal of Meteorology, vol. 17, No. 5, Oct.
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in the Intensity of the Zonal Circulation of the Atmosphere
and the Displacements of the Semi-permanent Centers of
Action,” Journal of Marine Research, vol. 2, No. 1, 1939,
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in Certain Types of Oceanic and Atmospheric Waves,” Journal
of Meteorology, vol. 2, No. 4, Dec. 1945, pp. 187-204.
10. N. P. Sellick, “Equatorial Circulations,” Quarterly Journal of
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12. T. C. Yeh, “On Energy Dispersion in the Atmosphere,”
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pp. 607-609.
126-164.
1960, pp. 484-48s.
pp. 38-55.
pp. 89-94.
[Received J u n e 7 , 1965; revised August 17, 19651